## Seminars and Colloquia by Series

### The Grand Arc Graph -- A "curve graph" for infinite-type surfaces

Series
Geometry Topology Seminar
Time
Monday, March 14, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Assaf Bar-NatanUniversity of Toronto

In this talk, I will be defining the grand arc graph for infinite-type surfaces. This simplicial graph is motivated by the works of Fanoni-Ghaswala-McLeay, Bavard, and Bavard-Walker to define an infinite-type analogue of the curve graph. As in these earlier works, the grand arc graph is connected, (oftentimes) infinite-diameter, and (sometimes) delta hyperbolic. Moreover, the mapping class group acts on it by isometries, and the action is continuous on the visible boundary. If there's time, this talk will degenerate into open speculation about what the boundary looks like and what we can do with it.

### An invariant for families of contact structures in monopole Floer homology

Series
Geometry Topology Seminar
Time
Monday, March 7, 2022 - 14:00 for
Location
Skiles 006
Speaker
Juan Muñoz-EchánizColumbia University

The contact invariant, introduced by Kronheimer and Mrowka,
is an element in the monopole Floer homology of a 3-manifold which is
canonically attached to a contact structure. I will describe an
application of monopole Floer homology and the contact invariant to
study the topology of spaces of contact structures and
contactomorphisms on 3-manifolds. The main new tool is a version of
the contact invariant for families of contact structures.

### Finite-order mapping classes of del Pezzo surfaces

Series
Geometry Topology Seminar
Time
Monday, February 28, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Seraphina LeeUniversity of Chicago

Let $M$ be the underlying smooth $4$-manifold of a degree $d$ del Pezzo surface. In this talk, we will discuss two related results about finite subgroups of the mapping class group $\text{Mod}(M) := \pi_0(\text{Homeo}^+(M))$. A motivating question for both results is the Nielsen realization problem for $M$: which finite subgroups $G$ of $\text{Mod}(M)$ have lifts to $\text{Diff}^+(M) \leq \text{Homeo}^+(M)$ under the quotient map $\pi: \text{Homeo}^+(M) \to \text{Mod}(M)$? For del Pezzo surfaces $M$ of degree $d \geq 7$, we will give a complete classification of such finite subgroups. Furthermore, we will give a classification of, and a structure theorem for, all involutions in $\text{Mod}(M)$ for all del Pezzo surfaces $M$. This yields a positive solution to the Nielsen realization problem for involutions on $M$ and a connection to Bertini's classification of birational involutions of $\mathbb{CP}^2$ (up to conjugation by birational automorphisms of $\mathbb{CP}^2$).

### Symplectic Lefschetz fibrations, rational blowdowns, and small exotic 4-manifolds

Series
Geometry Topology Seminar
Time
Monday, February 21, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Jonathan SimoneGeorgia Tech

Fiber sums and the rational blowdown have been very useful tools in constructing smooth, closed, oriented 4-manifolds. Applying these tools to genus g>1 Lefschetz fibrations with clustered nodal fibers, we will construct symplectic Lefschetz fibrations realizing all the lattice points in the symplectic geography plane below the Noether line, providing a symplectic extension of classical works populating the complex geography plane with holomorphic Lefschetz fibrations. Moreover, Lefschetz fibrations with certain clustered nodal fibers provide rational blowdown configurations that yield new constructions of small symplectic exotic 4-manifolds. We will present an example of a construction of a minimal symplectic exotic CP^2#-5CP^2 through this procedure applied to a genus-3 fibration. This work is joint with Inanc Baykur and Mustafa Korkmaz.

### Abelian cycles in the homology of the Torelli group

Series
Geometry Topology Seminar
Time
Monday, February 14, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Online
Speaker
Erik LindellUniversity of Stockholm

The mapping class group of a compact and orientable surface of genus g has an important subgroup called the Torelli group, which is the kernel of the action on the homology of the surface. In this talk we will discuss the stable rational homology of the Torelli group of a surface with a boundary component, about which very little is known in general. These homology groups are representations of the arithmetic group Sp_{2g}(Z) and we study them using an Sp_{2g}(Z)-equivariant map induced on homology by the so-called Johnson homomorphism. The image of this map is a finite dimensional and algebraic representation of Sp_{2g}(Z). By considering a type of homology classes called abelian cycles, which are easy to write down for Torelli groups and for which we can derive an explicit formula for the map in question, we may use classical representation theory of symplectic groups to describe a large part of the image.

### Stable commutator length on big mapping class groups

Series
Geometry Topology Seminar
Time
Monday, February 7, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Speaker
Elizabeth FieldUniversity of Utah

The stable commutator length function measures the growth rate of the commutator length of powers of elements in the commutator subgroup of a group. In this talk, we will discuss the stable commutator length function on the mapping class groups of infinite-type surfaces which satisfy a certain topological characterization. In particular, we will show that stable commutator length is a continuous function on these big mapping class groups, as well as that the commutator subgroups of these big mapping class groups are both open and closed. Along the way to proving our main results, we will discuss certain topological properties of a class of infinite-type surfaces and their end spaces which may be of independent interest. This talk represents joint work with Priyam Patel and Alexander Rasmussen.

### Stein property of complex-hyperbolic Kleinian groups

Series
Geometry Topology Seminar
Time
Monday, January 31, 2022 - 14:00 for 1 hour (actually 50 minutes)
Location
Online
Speaker

Let M be a complex-hyperbolic n-manifold, i.e. a quotient of the complex-hyperbolic n-space $\mathbb{H}^n_\mathbb{C}$ by a torsion-free discrete group of isometries, $\Gamma = \pi_1(M)$. Suppose that M is  convex-cocompact, i.e. the convex core of M is a nonempty compact subset. In this talk, we will discuss a sufficient condition on $\Gamma$ in terms of the growth-rate of its orbits in $\mathbb{H}^n_\mathbb{C}$ for which M is a Stein manifold. We will also talk about some interesting questions related to this result. This is a joint work with Misha Kapovich.

https://bluejeans.com/196544719/9518

### The diffeomorphism group of a 4-manifold

Series
Geometry Topology Seminar
Time
Monday, January 24, 2022 - 14:00 for
Location
Online (Zoom)
Speaker
Danny RubermanBrandeis University

Associated to a smooth n-dimensional manifold are two infinite-dimensional groups: the group of homeomorphisms Homeo(M), and the group of diffeomorphisms, Diff(M). For manifolds of dimension greater than 4, the topology of these groups has been intensively studied since the 1950s. For instance, Milnor’s discovery of exotic 7-spheres immediately shows that there are distinct path components of the diffeomorphism group of the 6-sphere that are connected in its homeomorphism group.  The lowest dimension for such classical phenomena is 5.

I will discuss recent joint work with Dave Auckly about these groups in dimension 4. For each n, we construct a simply connected 4-manifold Z and an infinite subgroup of the nth homotopy group of Diff(Z) that lies in the kernel of the natural map to the corresponding homotopy group of Homeo(Z). These elements are detected by (n+1)—parameter gauge theory. The construction uses a topological technique.  I’ll mention some other applications to embeddings of surfaces and 3-manifolds in 4-manifolds.

### Applications of contact geometry to 3-dimensional Anosov dynamics

Series
Geometry Topology Seminar
Time
Monday, November 29, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
Online (also Skiles 006)
Speaker
Federico SalmoiraghiTechnion

Anosov flows are an important class of dynamical systems due to their ergodic properties and structural stability. Geometrically, they are defined by two transverse invariant foliations with expanding and contracting behaviors. Much of our understanding of the structure of an Anosov flow relies on the study of the leaves space of the invariant foliations. In this talk we adopt a different approach: in the early 90s Mitsumatsu first noticed that and Anosov vector field also belongs to the intersection of two transverse contact structures rotating towards each other. After giving the necessary background I will use this point of view to address questions in surgery theory on Anosov flows and contact structures.

### Detection results in link Floer homology

Series
Geometry Topology Seminar
Time
Monday, November 15, 2021 - 14:00 for 1 hour (actually 50 minutes)
Location
Skiles 006
Speaker
Subhankar DeyUniverity of Alabama

In this talk I will briefly describe link Floer homology toolbox and its usefulness. Then I will show how link Floer homology can detect links with small ranks, using a rank bound for fibered links by generalizing an existing result for knots. I will also show that stronger detection results can be obtained as the knot Floer homology can be shown to detect T(2,8) and T(2,10), and that link Floer homology detects (2,2n)-cables of trefoil and figure eight knot. This talk is based on a joint work with Fraser Binns (Boston College).